Expected Mean Square (EMS) is utilized to determine the correct analysis statistics, which is the F-Statistics. In a design with all fixed factors, only one random error is coming from the natural variations in the experimental units. Therefore, the divisor for the F-Statistics is used as the experimental errors due to the natural variations in the experimental units, which can be determined without developing the EMS table. However, this simplest case will help us understand the process in the development of the correct EMS table, and therefore, the correct F-Statistics for the analysis. The rules in developing the EMS Table are explained using three fixed factors, A, B, and C. A model consisting of three factors, A, B, and C is written as in Equation 3.

Note. Where everything follows the usual notations such as “a” is the number of levels of the factor “A.” Alpha (α) is the effect due to the factor A and “i” indicates the i^th level of factor A.

Equation 3

While the variance for the random factor/effect is written as the sigma square with the associated subscript, the mean sum of square is used as the variance for the fixed effect factor. For example, the variance for A is written as

EMS Rule #1

Write all of the factors, factor types, levels, replications, factorial effects, their variances, and subscripts as in the table below.

EMS Rule #2

Insert the level/replication if the row subscript(s) does not contain the column subscript.

EMS Rule #3

Insert a “1” if the row subscript(s) contains the column subscript but dead (nested). [Replicates are always nested within the treatment combinations (=(ijk)l). B(A) notation is used when B is nested in A.]

EMS Rule #4

Unrestricted Model

(Fixed and random factors mixed interaction term is assumed random).

If the row subscript(s) contains the column subscript, insert a

1. “0” when both subscripts are fixed

2. “1” when at least one subscript is random

Restricted Model

(Fixed and random factors mixed interaction term is NOT assumed random).

If the row subscript(s) contains the column subscript, insert a

1. “0” when the column subscript is fixed

2. “1” when the column subscript is random

Note. For a fixed effect model, there is no mixed interaction (fixed and random) term. Therefore, no restriction is necessary and by default, it is an unrestricted model.

EMS Rule #5

1. Hide/mask/delete the column (s) for which the EMS is to be determined. For example, to find EMS(BC), B and C columns are hidden/masked.

2. In the Expected Mean Square column, write the results by multiplying all the inserted values in a row and the associated variance of that row. For example, the Expected Mean Square component for B effect can be found as follows.

The Expected Mean Square component for BC effect can be written as an .

EMS Rule #6

To find the EMS for BC interaction term, add only the components which contain both B and C factors together. For example, neither B row nor C row contains both B and C together. Therefore, B and C rows should NOT be added to find the EMS for the BC interaction term. Both BC and the Error rows contain both B and C together. Therefore, BC and the Error rows are added to find the Expected Mean Square for BC.

Similarly, EMS for all other effects can be determined (Table 1). The correct divisor for the F-Statistics is the random error associated with the expected mean square for the respective effect (Table 1).

Table 1.

The EMS Table for A, B, and C Fixed Effect Model